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Statement

Context (p. 849). Lorentz asked whether some sequence $b_j$ with $N(b_j,x)<c_{10}x^{1/2}$ has every large integer of the form $k^2+b_j$. The paper notes that Lorentz's bound (1), or the method of Theorem 1, gives only $N(b_j,x)<c_{11}x^{1/2}\log x$.

Remark (p. 853, unnumbered, added after the paper was finished). There is a sequence $b_1<b_2<\cdots$ with $N(b_j,x)<c_{10}x^{1/2}$ such that every large integer is of the form $l^2+b_j$. The paper calls this easy and says it suffices to take as the $b$'s the integers of the intervals

$$2^k<b<2^k+4\cdot2^{k/2},\qquad k=1,2,\ldots.$$

Analogue for $k$-th powers (p. 853). The paper states that an analogous example gives a sequence $b_1<b_2<\cdots$ with $N(b_j,x)<c_kx^{1-1/k}$ such that every sufficiently large integer is of the form $l^k+b_j$.

Source. P. Erdős, Some results on additive number theory, Proc. Amer. Math. Soc. 5 (1954), 847-853: Lorentz's question on p. 849, the remark and its analogue on p. 853. The edition read is identified on the source card.

Read depth. Claims checked: the remark and its analogue were read clause by clause on the printed page. The paper gives the construction for squares without a proof and no construction for $k$-th powers. Nothing here is independently reviewed.

Proof pointer

Page 853. The paper gives only the intervals above, calls the verification easy, and gives no proof; for $k$-th powers it gives no construction.

Dependencies

None beyond the elementary spacing of squares.

Bears on

• Problem 33: the problem asks for the smallest possible $\limsup\lvert A\cap\{1,\ldots,N\}\rvert/N^{1/2}$ over sets $A$ with every large integer of the form $n^2+a$, and whether the liminf exceeds $1$. The remark gives such a set with $\lvert A\cap\{1,\ldots,N\}\rvert<c_{10}N^{1/2}$, so the smallest limsup is finite; the paper names no value for $c_{10}$, and the remark determines neither the smallest limsup nor the liminf question.